Financial Statistics and Mathematical Finance Methods, Models and Applications. Ansgar Steland

Transcription

1 Financial Statistics and Mathematical Finance Methods, Models and Applications Ansgar Steland

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3 Financial Statistics and Mathematical Finance

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5 Financial Statistics and Mathematical Finance Methods, Models and Applications Ansgar Steland Institute for Statistics and Economics RWTH Aachen University, Germany

6 This edition first published John Wiley & Sons, Ltd Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Library of Congress Cataloging-in-Publication Data Steland, Ansgar. Financial statistics and mathematical finance : methods, models and applications / Ansgar Steland. Includes bibliographical references and index. ISBN Business mathematics. 2. Calculus. I. Title. HF5691.S dc A catalogue record for this book is available from the British Library. ISBN: Set in 10/12pt Times by Thomson Digital, Noida, India.

7 Contents Preface Acknowledgements xi xv 1 Elementary financial calculus Motivating examples Cashflows, interest rates, prices and returns Bonds and the term structure of interest rates Asset returns Some basic models for asset prices Elementary statistical analysis of returns Measuring location Measuring dispersion and risk Measuring skewness and kurtosis Estimation of the distribution Testing for normality Financial instruments Contingent claims Spot contracts and forwards Futures contracts Options Barrier options Financial engineering A primer on option pricing The no-arbitrage principle Risk-neutral evaluation Hedging and replication Nonexistence of a risk-neutral measure The Black Scholes pricing formula The Greeks Calibration, implied volatility and the smile Option prices and the risk-neutral density Notes and further reading 43 References 43 2 Arbitrage theory for the one-period model Definitions and preliminaries Linear pricing measures More on arbitrage 50

8 vi CONTENTS 2.4 Separation theorems in R n No-arbitrage and martingale measures Arbitrage-free pricing of contingent claims Construction of martingale measures: general case Complete financial markets Notes and further reading 76 References 76 3 Financial models in discrete time Adapted stochastic processes in discrete time Martingales and martingale differences The martingale transformation Stopping times, optional sampling and a maximal inequality Extensions to R d Stationarity Weak and strict stationarity Linear processes and ARMA models Linear processes and the lag operator Inversion AR(p) and AR( ) processes ARMA processes The frequency domain The spectrum The periodogram Estimation of ARMA processes (G)ARCH models Long-memory series Fractional differences Fractionally integrated processes Notes and further reading 144 References Arbitrage theory for the multiperiod model Definitions and preliminaries Self-financing trading strategies No-arbitrage and martingale measures European claims on arbitrage-free markets The martingale representation theorem in discrete time The Cox Ross Rubinstein binomial model The Black Scholes formula American options and contingent claims Arbitrage-free pricing and the optimal exercise strategy Pricing american options using binomial trees Notes and further reading 175 References 175

9 CONTENTS vii 5 Brownian motion and related processes in continuous time Preliminaries Brownian motion Definition and basic properties Brownian motion and the central limit theorem Path properties Brownian motion in higher dimensions Continuity and differentiability Self-similarity and fractional Brownian motion Counting processes The poisson process The compound poisson process Lévy processes Notes and further reading 201 References Itô Calculus Total and quadratic variation Stochastic Stieltjes integration The Itô integral Quadratic covariation Itô s formula Itô processes Diffusion processes and ergodicity Numerical approximations and statistical estimation Notes and further reading 239 References The Black Scholes model The model and first properties Girsanov s theorem Equivalent martingale measure Arbitrage-free pricing and hedging claims The delta hedge Time-dependent volatility The generalized Black Scholes model Notes and further reading 261 References Limit theory for discrete-time processes Limit theorems for correlated time series A regression model for financial time series Least squares estimation Limit theorems for martingale difference 278

10 viii CONTENTS 8.4 Asymptotics Density estimation and nonparametric regression Multivariate density estimation Nonparametric regression The CLT for linear processes Mixing processes Mixing coefficients Inequalities Limit theorems for mixing processes Notes and further reading 323 References Special topics Copulas and the 2008 financial crisis Copulas The financial crisis Models for credit defaults and CDOs Local Linear nonparametric regression Applications in finance: estimation of martingale measures and Itô diffusions Method and asymptotics Change-point detection and monitoring Offline detection Online detection Unit roots and random walk The OLS estimator in the stationary AR(1) model Nonparametric definitions for the degree of integration The Dickey Fuller test Detecting unit roots and stationarity Notes and further reading 381 References 382 Appendix A 385 A.1 (Stochastic) Landau symbols 385 A.2 Bochner s lemma 387 A.3 Conditional expectation 387 A.4 Inequalities 388 A.5 Random series 389 A.6 Local martingales in discrete time 389 Appendix B Weak convergence and central limit theorems 391 B.1 Convergence in distribution 391 B.2 Weak convergence 392

11 CONTENTS ix B.3 Prohorov s theorem 398 B.4 Sufficient criteria 399 B.5 More on Skorohod spaces 401 B.6 Central limit theorems for martingale differences 402 B.7 Functional central limit theorems 403 B.8 Strong approximations 405 References 407 Index 409

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13 Preface This textbook intends to provide a careful and comprehensive introduction to some of the most important mathematical topics required for a thorough understanding of financial markets and the quantitative methods used there. For this reason, the book covers mathematical finance in the narrow sense, that is, arbitrage theory for pricing contingent claims such as options and the related mathematical machinery, as well as statistical models and methods to analyze data from financial markets. These areas evolved more or less separate from each other and the lack of material that covers both was a major motivation for me to work out the present textbook. Thus, I wrote a book that I would have liked when taking up the subject. It addresses master and Ph.D. students as well as researchers and practitioners interested in a comprehensive presentation of both areas, although many chapters can also be studied by Bachelor students who have passed introductory courses in probability calculus and statistics. Apart from a couple of exceptions, all results are proved in detail, although usually not in their most general form. Given the plethora of notions, concepts, models and methods and the resulting inherent complexity, particularly those coming to the subject for the first time can acquire a thorough understanding more quickly, if they can easily follow the derivations and calculations. For this reason, the mathematical formalism and notation is kept as elementary as possible. Each chapter closes with notes and comments on selected references, which may complement the presented material or are good starting points for further studies. Chapter 1 starts with a basic introduction to important notions: financial instruments such as options and derivatives and related elementary methods. However, derivations are usually not given in order to focus on ideas, principles and basic results. It sets the scene for the following chapters and introduces the required financial slang. Cash flows, discounting and the term structure of interest rates are studied at an elementary level. The return over a given period of time, for assets usually a day, represents the most important economic object of interest in finance, as prices can be reconstructed from returns and investments are judged by comparing their return. Statistical measures for their location, dispersion and skewness have important economic interpretations, and the relevant statistical approaches to estimate them are carefully introduced. Measuring the risk associated with an investment requires being aware of the properties of related statistical estimates. For example, volatility is primarily related to the standard deviation and value-at-risk, by definition, requires the study of quantiles and their statistical estimation. The first chapter closes with a primer on option pricing, which introduces the most important notions of the field of mathematical finance in the narrow sense, namely the principle of no-arbitrage, the principle of risk-neutral pricing and the relation of those notions to probability calculus, particularly to the existence of an equivalent martingale measure. Indeed, these basic concepts and a couple of fundamental insights can be understood by studying them in the most elementary form or simply by examples. Chapter 2 then discusses arbitrage theory and the pricing of contingent claims within a one-period model. At time 0 one sets up a portfolio and at time 1 we look at the result. Within this simple framework, the basic results discussed in Chapter 1 are treated with mathematical rigor and extended from a finite probability space, where only a finite number of scenarios

14 xii PREFACE can occur, to a general underlying probability space that models the real financial market. Mathematical separation theorems, which tell us how one can separate a given point from convex sets, are applied in order to establish the equivalence of the exclusion of arbitrage opportunities and the existence of an equivalent martingale measure. For this reason, those separation theorems are explicitly proved. The construction of equivalent martingale measures based on the Esscher transform is discussed as well. Chapter 3 provides a careful introduction to stochastic processes in discrete time (time series), covering martingales, martingale differences, linear processes, ARMA and GARCH processes as well as long-memory series. The notion of a martingale is fundamental for mathematical finance, as one of the key results asserts that in any financial market that excludes arbitrage, there exists a probability measure such that the discounted price series of a risky asset forms a martingale and the pricing of contingent claims can be done by risk-neutral pricing under that measure. These key insights allow us to apply the elaborated mathematical theory of martingales. However, the treatment in Chapter 3 is restricted to the most important findings of that theory, which are really used later. Taking first-order differences of a martingale leads naturally to martingale difference sequences, which form whitenoise processes and are a common replacement for the unrealistic i.i.d. error terms in stochastic models for financial data and, more generally, economic data. A key empirical insight of the statistical analysis of financial return series is that they can often be assumed to be uncorrelated, but they are usually not independent. However, other series may exhibit substantial serial dependence that has to be taken into account. Appropriate parametric classes of time-series models are ARMA processes, which belong to the more general and infinite-dimensional class of linear processes. Basic approaches to estimate autocovariance functions and the parameters of ARMA models are discussed. Many financial series exhibit the phenomenon of conditional heteroscedasticity, which has given rise to the class of (G)ARCH models. Lastly, fractional differences and longmemory processes are introduced. Chapter 4 discusses in detail arbitrage theory in a discrete-time multiperiod model. Here, trading is allowed at a finite number of time points and at each time point the trading strategy can be updated using all available information on market prices. Using the martingale theory in discrete time studied in Chapter 3, it allows us to investigate the pricing of options and other derivatives on arbitrage-free financial markets. The Cox Ross Rubinstein binomial model is studied in greater detail, since it is a standard tool in practice and also provides the basis to derive the famous Black Scholes pricing formula for a European call. In addition, the pricing of American claims is studied, which requires some more advanced results from the theory of optimal stopping. Chapter 5 introduces the reader to stochastic processes in continuous time. Brownian motion will be the random source that governs the price processes of our financial market model in continuous time. Nevertheless, to keep the chapter concise, the presentation of Brownian motion is limited to its definition and the most important properties. Brownian motion has puzzling properties such as continuous paths that are nowhere differentiable or of bounded variation. Advanced models also incorporate fractional Brownian motion and Lévy processes, respectively. Lévy processes inherit independent increments but allow for nonnormal distributions of those increments including heavy tails and jump. Fractional Brownian motion is a Gaussian process as is Brownian motion, but it allows for long-range dependent increments where temporal correlations die out very slowly. Chapter 6 treats the theory of stochastic integration. Assuming that the reader is familiar with integration in the sense of Riemann or Lebesgue, we start with a discussion of stochastic

15 PREFACE xiii Riemann Stieltjes (RS) integrals, a straightforward generalization of the Riemann integral. The related calculus is relatively easy and provides a good preparation for the Itô integral. It is also worth mentioning that the stochastic RS-integral definitely suffices to study many issues arising in statistics. However, the problems arising in mathematical finance cannot be treated without the Itô integral. The key observation is that the change of the value of position x(t) = x t in a stock at time t over the period [t, t + δ] is, of course, given by x t δp t, where δp t = P t+δ P t. Aggregating those changes over n successive time intervals [iδ, (i + 1)δ], i = 0,...,n 1, in order to determine the terminal value, results in the sum n 1 i=0 x(iδ)δp iδ. Now taking the limit δ 0 leads to an integral x s dp s with respect to the stock price, which cannot be defined in the Stieltjes sense, if the stock price is not of bounded variation. Here the Itô integral comes into play. A rich class of processes are Itô processes and the famous Itô formula asserts that smooth functions of Itô processes again yield Itô processes, whose representation as an Itô process can be explicitly calculated. Further, ergodic diffusion processes as an important class of Itô processes are introduced as well as Euler s numerical approximation scheme, which also provides the common basis for statistical estimation and inference of discretely sampled ergodic diffusions. Chapter 7 presents the Black Scholes model, the mathematically idealized model to price derivatives which is still the benchmark continuous-time model in practice. Here one may either invest in a risky stock or deposit money in a bank account that pays a fixed interest. The Itô calculus of Chapter 6 provides the theoretical basis to develop the mathematical arbitrage theory in continuous time. The classic Black Scholes model assumes that the volatility of the stock price is constant with respect to time, which is too restrictive in practice. Thus, we briefly discuss the required changes when the volatility is time dependent but deterministic. Finally, the generalized Black Scholes model allows the interest rate of the risk-less instrument to be random as well as dependent on time, thus covering the realistic situation that money not invested in stocks is used to buy, for example, AAA-rated government bonds. Chapter 8 studies the asymptotic limit theory for discrete-time processes as required to construct and investigate present-day methods for decision making; that is, procedures for estimation, inference as well as model checking, using financial data in the widest sense (returns, indexes, prices, risk measures, etc.). The limit theorems, partly presented along the way when needed to develop methodologies, cover laws of large numbers and central limit theorems for martingale differences, linear processes as well as mixing processes. The methods discussed in greater detail cover the multiple linear regression with stochastic regressors, nonparametric density estimation, nonparametric regression and the estimation of autocovariances and the long-run variance. Those statistical tools are ubiquitous in the analysis of financial data. Chapter 9 discusses some selected topics. Copulas have become an important tool for modeling high-dimensional distributions with powerful as well as dangerous applications in the pricing of financial instruments related to credits and defaults. As a matter of fact, these played an unlucky role in the 2008 financial crisis when a simplistic pricing model was applied to large-scale pricing of credit default obligations. For this reason, some of the major developments leading to the crisis are briefly reviewed, revealing the inherent complexity of financial markets as well as the need for sophisticated mathematical models and their application. Local polynomial estimation is studied in greater detail, since it has important applications to many problems arising in finance such as the estimation of risk-neutral densities conditional volatility or discretely observed diffusion processes. The asymptotic normality can be based on a powerful reduction principle: A (joint) smoothing central limit theorem for the innovation process {ɛ t } and a derived process involving the regressors automatically